Nick almost had a parental role with Nate when they were teens, so it probably feels great and relieving to know that Nate is set up for life. It probably makes it feel even better for Nick to know that he was the one who discovered martial arts and guided Nate towards them and away from gangs and dangerous drugs.
My guess too is that a lot of casual fans first learned about Diaz from the Conor fight, and they couldn't reliably tell the difference between Nate and Nick. I also bet there is even a portion of casual fans who think that Nate was the guy who fought GSP and Anderson Silva.